for their pioneering analysis of equilibria in the theory of
non-cooperative games.

Games as the Foundation for Understanding Complex Economic
Issues
Game theory emanates from studies of games such as chess or
poker. Everyone knows that in these games, players have to think
ahead - devise a strategy based on expected countermoves from the
other player(s). Such strategic interaction also characterizes
many economic situations, and game theory has therefore proved to
be very useful in economic analysis.

The foundations for using game theory in economics were
introduced in a monumental study by John von Neumann and Oskar
Morgenstern entitled Theory of Games and Economic Behavior
(1944). Today, 50 years later, game theory has become a dominant
tool for analyzing economic issues. In particular,
non-cooperative game theory, i.e., the branch of game theory
which excludes binding agreements, has had great impact on
economic research. The principal aspect of this theory is the
concept of equilibrium, which is used to make predictions about
the outcome of strategic interaction. John F. Nash, Reinhard
Selten and John C. Harsanyi are three researchers who have made
eminent contributions to this type of equilibrium analysis.

John F. Nash introduced the distinction between
cooperative games, in which binding agreements can be made, and
non-cooperative games, where binding agreements are not feasible.
Nash developed an equilibrium concept for non-cooperative games
that later came to be called Nash equilibrium.

Reinhard Selten was the first to refine the Nash
equilibrium concept for analyzing dynamic strategic interaction.
He has also applied these refined concepts to analyses of
competition with only a few sellers.

John C. Harsanyi showed how games of incomplete
information can be analyzed, thereby providing a theoretical
foundation for a lively field of research - the economics of
information - which focuses on strategic situations where
different agents do not know each others' objectives.

Strategic Interaction
Game theory is a mathematical method for analyzing strategic
interaction. Many classical analyses in economics presuppose
such a large number of agents that each of them can disregard the
others' reactions to their own decision. In many cases, this
assumption is a good description of reality, but in other cases
it is misleading. When a few firms dominate a market, when
countries have to make an agreement on trade policy or
environmental policy, when parties on the labor market negotiate
about wages, and when a government deregulates a market,
privatizes companies or pursues economic policy, each agent in
question has to consider other agents' reactions and expectations
regarding their own decisions, i.e., strategic interaction.

As far back as the early nineteenth century, beginning with
Auguste Cournot in 1838, economists have developed methods for
studying strategic interaction. But these methods focused on
specific situations and, for a long time, no overall method
existed. The game-theoretic approach now offers a general
toolbox for analyzing strategic interaction.

Game Theory
Whereas mathematical probability theory ensued from the study of
pure gambling without strategic interaction, games such as chess,
cards, etc. became the basis of game theory. The latter are
characterized by strategic interaction in the sense that the
players are individuals who think rationally. In the early 1900s,
mathematicians such as Zermelo, Borel and von Neumann had already
begun to study mathematical formulations of games. It was not
until the economist Oskar Morgenstern met the mathematician John
von Neumann in 1939 that a plan originated to develop game theory
so that it could be used in economic analysis.

The most important ideas set forth by von Neumann and Morgenstern
in the present context may be found in their analysis of
two-person zero-sum games. In a zero-sum game, the gains of one
player are equal to the losses of the other player. As early as
1928, von Neumann introduced the minimax solution for a
two-person zero-sum game. According to the minimax solution, each
player tries to maximize his gain in the outcome which is most
disadvantageous to him (where the worst outcome is determined by
his opponent's choice of strategy). By means of such a strategy,
each player can guarantee himself a minimum gain. Of course, it
is not certain that the players' choices of strategy will be
consistent with each other. von Neumann was able to show,
however, that there is always a minimax solution, i.e., a
consistent solution, if so-called mixed strategies are
introduced. A mixed strategy is a probability distribution of a
player's available strategies, whereby a player is assumed to
choose a certain "pure" strategy with some probability.

John F. Nash John Nash arrived at Princeton
University in 1948 as a young doctoral student in mathematics.
The results of his studies are reported in his doctoral
dissertation entitled Non-cooperative Games (1950). The
thesis gave rise to Equilibrium Points in n-person Games
(Proceedings of the National Academy of Sciences of the USA
1950), and to an article entitled Non-cooperative Games, (Annals
of Mathematics 1951).

In his dissertation, Nash introduced the distinction between
cooperative and non-cooperative games. His most important
contribution to the theory of non-cooperative games was to
formulate a universal solution concept with an arbitrary number
of players and arbitrary preferences, i.e., not solely for
two-person zero-sum games. This solution concept later came to be
called Nash equilibrium. In a Nash equilibrium, all of the
players' expectations are fulfilled and their chosen strategies
are optimal. Nash proposed two interpretations of the equilibrium
concept: one based on rationality and the other on statistical
populations. According to the rationalistic interpretation, the
players are perceived as rational and they have complete
information about the structure of the game, including all of the
players' preferences regarding possible outcomes, where this
information is common knowledge. Since all players have complete
information about each others' strategic alternatives and
preferences, they can also compute each others' optimal choice of
strategy for each set of expectations. If all of the players
expect the same Nash equilibrium, then there are no incentives
for anyone to change his strategy. Nash's second interpretation -
in terms of statistical populations - is useful in so-called
evolutionary games. This type of game has also been developed in
biology in order to understand how the principles of natural
selection operate in strategic interaction within and among
species. Moreover, Nash showed that for every game with a finite
number of players, there exists an equilibrium in mixed
strategies.

Many interesting economic issues, such as the analysis of
oligopoly, originate in non-cooperative games. In general, firms
cannot enter into binding contracts regarding restrictive trade
practices because such agreements are contrary to trade
legislation. Correspondingly, the interaction among a government,
special interest groups and the general public concerning, for
instance, the design of tax policy is regarded as a
non-cooperative game. Nash equilibrium has become a standard tool
in almost all areas of economic theory. The most obvious is
perhaps the study of competition between firms in the theory of
industrial organization. But the concept has also been used in
macroeconomic theory for economic policy, environmental and
resource economics, foreign trade theory, the economics of
information, etc. in order to improve our understanding of
complex strategic interactions. Non-cooperative game theory has
also generated new research areas. For example, in combination
with the theory of repeated games, non-cooperative equilibrium
concepts have been used successfully to explain the development
of institutions and social norms. Despite its usefulness, there
are problems associated with the concept of Nash equilibrium. If
a game has several Nash equilibria, the equilibrium criterion
cannot be used immediately to predict the outcome of the game.
This has brought about the development of so-called refinements
of the Nash equilibrium concept. Another problem is that when
interpreted in terms of rationality, the equilibrium concept
presupposes that each player has complete information about the
other players' situation. It was precisely these two problems
that Selten and Harsanyi undertook to solve in their
contributions.

Reinhard Selten The problem of numerous
non-cooperative equilibria has generated a research program aimed
at eliminating "uninteresting" Nash equilibria. The principal
idea has been to use stronger conditions not only to reduce the
number of possible equilibria, but also to avoid equilibria which
are unreasonable in economic terms. By introducing the concept of
subgame perfection, Selten provided the foundation for a
systematic endeavor in Spieltheoretische Behandlung eines
Oligopolmodellsmit Nachfrageträgheit, (Zeitschrift
für die Gesamte Staatswissenschaft 121, 301-24 and 667-89,
1965).

An example might help to explain this concept. Imagine a monopoly
market where a potential competitor is deterred by threats of a
price war. This may well be a Nash equilibrium - if the
competitor takes the threat seriously, then it is optimal to stay
out of the market - and the threat is of no cost to the
monopolist because it is not carried out. But the threat is not
credible if the monopolist faces high costs in a price war. A
potential competitor who realizes this will establish himself on
the market and the monopolist, confronted with fait
accompli, will not start a price war. This is also a Nash
equilibrium. In addition, however, it fulfills Selten's
requirement of subgame perfection, which thus implies systematic
formalization of the requirement that only credible threats
should be taken into account.

Selten's subgame perfection has direct significance in
discussions of credibility in economic policy, the analysis of
oligopoly, the economics of information, etc. It is the most
fundamental refinement of Nash equilibrium. Nevertheless, there
are situations where not even the requirement of subgame
perfection is sufficient. This prompted Selten to introduce a
further refinement, usually called the "trembling-hand"
equilibrium, in Reexamination of the Perfectness Concept for
Equilibrium Points in Extensive Games (International Journal
of Game Theory 4, 25-55, 1975). The analysis assumes that each
player presupposes a small probability that a mistake will occur,
that someone's hand will tremble. A Nash equilibrium in a game is
"trembling-hand perfect" if it is robust with respect to small
probabilities of such mistakes. This and closely related
concepts, such as sequential equilibrium (Kreps and Wilson,
1982), have turned out to be very fruitful in several areas,
including the theory of industrial organization and macroeconomic
theory for economic policy.

John C. Harsanyi In games with complete information,
all of the players know the other players' preferences, whereas
they wholly or partially lack this knowledge in games with
incomplete information. Since the rationalistic interpretation of
Nash equilibrium is based on the assumption that the players know
each others' preferences, no methods had been available for
analyzing games with incomplete information, despite the fact
that such games best reflect many strategic interactions in the
real world.

This situation changed radically in 1967-68 when John Harsanyi
published three articles entitled Games with Incomplete
Information Played by Bayesian Players, (Management Science
14, 159-82, 320-34 and 486-502). Harsanyi's approach to games
with incomplete information may be viewed as the foundation for
nearly all economic analysis involving information, regardless of
whether it is asymmetric, completely private or public.

Harsanyi postulated that every player is one of several "types",
where each type corresponds to a set of possible preferences for
the player and a (subjective) probability distribution over the
other players' types. Every player in a game with incomplete
information chooses a strategy for each of his types. Under a
consistency requirement on the players' probability
distributions, Harsanyi showed that for every game with
incomplete information, there is an equivalent game with complete
information. In the jargon of game theory, he thereby transformed
games with incomplete information into games with imperfect
information. Such games can be handled with standard
methods.

An example of a situation with incomplete information is when
private firms and financial markets do not exactly know the
preferences of the central bank regarding the tradeoff between
inflation and unemployment. The central bank's policy for future
interest rates is therefore unknown. The interactions between the
formation of expectations and the policy of the central bank can
be analyzed using the technique introduced by Harsanyi. In the
most simple case, the central bank can be of two types, with
adherent probabilities: Either it is oriented towards fighting
inflation and thus prepared to pursue a restrictive policy with
high rates, or it will try to combat unemployment by means of
lower rates. Another example where similar methods can be applied
is regulation of a monopoly firm. What regulatory or contractual
solution will produce a desirable outcome when the regulator does
not have perfect knowledge about the firm's costs?

Other Contributions of the Laureates
In addition to his contributions to non-cooperative game theory,
John Nash has developed a basic solution for cooperative games,
usually referred to as Nash's bargaining solution, which has been
applied extensively in different branches of economic theory. He
also initiated a project that subsequently came to be called the
Nash program, a research program designed to base cooperative
game theory on results from non-cooperative game theory. In
addition to his prizewinning achievements, Reinhard Selten has
contributed powerful new insights regarding evolutionary games
and experimental game theory. John Harsanyi has also made
significant contributions to the foundations of welfare economics
and to the area on the boundary between economics and moral
philosophy. Harsanyi and Selten have worked closely together for
more than 20 years, sometimes in direct collaboration.

Through their contributions to equilibrium
analysis in non-cooperative game theory, the three laureates
constitute a natural combination: Nash provided the
foundations for the analysis, while Selten developed it
with respect to dynamics, and Harsanyi with respect to
incomplete information.